L3 Maximum and Minimum Values - Part 5
Completion requirements
Unit 3
Curve Sketching
Lesson 3: Maximum and Minimum Values
When the derivative is zero or infinite
Often, but not always, at the points of transition from increase to decrease or decrease to increase, the slope of a curve is zero or infinite; that is, the tangent line to the curve is either horizontal or vertical, as shown below.
However, the slope at a maximum or minimum is not necessarily zero or infinite. As shown in the graph below, the slope can also be undefined at a local minimum (or maximum).
As the cusp is approached from the left, the slope of the curve is not the same as when the cusp is approached from the right. As such, the derivative of the function at the cusp does not exist.
Additionally, determining that the slope (derivative) at a point is zero or infinite is not always sufficient to conclude a local maximum or minimum value exists, as shown below.
Often, but not always, at the points of transition from increase to decrease or decrease to increase, the slope of a curve is zero or infinite; that is, the tangent line to the curve is either horizontal or vertical, as shown below.
| Note: This graph represents a function that is not continuous.
As such, the function shown in this graph, with the vertical tangent, does not have a local minimum where the graph transitions from decreasing to increasing. |
Additionally, determining that the slope (derivative) at a point is zero or infinite is not always sufficient to conclude a local maximum or minimum value exists, as shown below.